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Briefly Noted

August 2005

Saber Elaydi suggests in his Preface to the third edition of An Introduction to Difference Equations that this textbook for “advanced undergraduate and beginning graduate [students]” be used for a two-semester course, following one of four options as a function of taste and interest: stability theory, asymptotic theory, oscillation theory, or control theory. The book seems to fit the bill perfectly. It is very well-written and thorough in its coverage of topics (which are fascinating and numerous, notwithstanding the author’s insistence that he is not interested in encyclopedic coverage). Additionally the book is full of good exercises at all levels replete with hints and answers (although Elaydi even goes so far at to sprinkle in an occasional unsolved problem!) and is rich in good examples. It is impossible not to admire Elaydi’s achievement in putting together a textbook of such quality.

Difference equations are of course to some extent analogous to differential equations and this parallel is already evident in the familiar appearance of the general solution of a homogeneous linear difference equation with constant coefficients: just as with homogeneous linear ordinary differential equations (also with constant coefficients, of course) one immediately goes over to an associated characteristic polynomial whose roots are intimately involved in the manufacture of the according general solution. In this connection Elaydi presents e.g. the example of the difference equation realizing the Fibonacci numbers, the upshot being that, qua general solution, the general Fibonacci number is expressible in terms of powers of the golden section and its conjugate, the roots of one of the most famous characteristic polynomials of all.

Proceeding to the non-homogeneous case we encounter further parallels with ordinary differential equations: the methods of undetermined coefficients (which is actually misspelled as “undetermind coefficeints” — an isolated singularity) and variation of parameters appear, soon to be followed by a study of the limiting behavior of solutions. Elaydi goes on to play with some nice examples, featuring an elegant treatment of the gambler’s ruin problem.

There is of course a thorough treatment of difference equations per se, culminating in a discussion of Markov chains. Then we encounter stability theory, including coverage of Lyapunov’s second method, and a discussion of Z-transforms and Volterra difference equations featuring a section titled, “The Z-transform Versus the Laplace Transform.”

After a chapter on oscillation theory, Elaydi hones in on the asymptotic behavior of difference equations in greater detail, with the focus falling on Poincaré’s Theorem and its extension by O. Perron. This is beautiful mathematics: even in the case of homogeneous linear difference equations with possibly non-constant coefficients, the general solution, x(n), generally has the property that the quotient x(n+1)/x(n) tends (as n → ∞) to a root of an associated (still “characteristic”) polynomial, with the case of constant coefficients a leitmotiv (as exemplified dramatically by, again, the Fibonacci numbers with F(n+1)/F(n)→ λ, the golden section).

Subsequently the theory of continued fractions is given a systematic going-over and Elaydi’s treatment of the connection between continued fractions and infinite series is particularly evocative: it even includes some very nice material on the Riemann ζ-function (if k is an integer ≥2, ζ(k) itself is rendered as a continued fraction with wonderfully suggestive numerators and denominators).

Thus, An Introduction to Difference Equations is a terrific book almost every page of which contains marvelous things. It will serve all the pedagogical purposes Elaydi delineates, even though the wealth of material in the book will often tempt the reader to go off in tangent or orthogonal directions at the risk of destroying the pace of the coverage. But wherever one ends up, it will be a trip well worth taking. [Michael Berg]

Exploratory Galois Theory is designed as a first undergraduate course on field and Galois theory, with a course in abstract algebra — groups and rings — as prerequisite. As a first intuitive approach to Galois theory, the book concentrates on the subfields of the complex numbers.

The first half of the book is dedicated to field theory: polynomial rings, roots, ring homomorphisms; algebraic numbers, field extensions, minimal polynomials; simple extensions, etc. The second half is dedicated to Galois theory: normal extensions and splitting fields; the Galois group and the Galois correspondence; resolvents, discriminants and computation of Galois groups. The last chapter of the book is dedicated to classical topics such as an introduction to Kummer theory and cyclic extensions; characteristic p and finite fields; ruler-and-compass constructions and solvability by radicals. Throughout the book, there are numerous sections which explain how to work with the previously described mathematical objects, using software: Maple and Mathematica (the text includes plenty of screenshots in which the reader can see how to type the needed expressions). For example, the reader learns how to define and factor polynomials, approximate complex roots, how to define and work with algebraic number fields, and of course, how to calculate Galois groups of polynomials and resolvents.

The goal of the author is, in his own words, "to develop Galois theory in as accessible a manner as possible for an undergraduate audience". The reviewer thinks that the goal was very nicely accomplished in this book, where a beautiful and comprehensive exposition of the abstract theory is greatly enhanced by the computational aspects with the help of software. The students who belong to the "calculator religion" will enormously benefit from the numerous hands-on examples and being able to work explicitly with fields and groups on the computer, while the more abstract-minded students will enjoy the excellent mathematical writing of the book. However, as mentioned earlier, the text is a basic introduction to Galois and field theory, mostly concentrating on subfields of the complex numbers, so, depending on the audience, the book's scope might be too narrow.

Finally, the reviewer would like to end this note with a personal concern. The undergraduate student (or at least an algebraically oriented student) will have to purchase books which cover groups, rings, fields, Galois theory and so on (other topics, even if not covered in class, might be handy for the student in the future). Should the instructor choose a couple of books which cover (some of) these topics or should the instructor pick a book which contains all of the previous topics (such as Abstract Algebra by Dummit and Foote)? [Álvaro Lozano Robledo]

When one picks up a book with a title like A First Graduate Course in Abstract Algebra, it is not clear what one will find inside the book. Different schools, and even different professors, seem to have vastly different ideas of what a course in Algebra should consist of, let alone what material belongs in the graduate, as opposed to the undergraduate curriculum. The new book of this title by W.J. Wickless starts at the very beginning of algebra. In fact, the first few chapters of the book cover the basics of group theory and ring theory and are mostly material that I would think of as undergraduate topics, albeit at a pace that would probably not be appropriate for undergraduates. While the book then discusses modules at a depth that I have never seen in an undergraduate textbook, the pace does not slow down as it does so. The book then quickly moves on to discuss vector spaces, fields, and Galois theory all at a level that I think of as more appropriate for an undergraduate text. It is only in the last several chapters that the book moves on to more advanced topics such as group extensions, noncommutative rings, and various structure theorems.

The book is well-written, and contains many nice exercises. However, there are lots of well-written introductory Algebra textbooks out there, and so it seems to me that a new book should bring something new to the table, and I am not convinced that Wickless does this. Mostly, I am not sure I understand who would find use for this book. It seems to me that many — if not most — graduate students would already have seen the majority of topics covered in the book, but I think that the pace of the book, as well as a lack of lots of good examples of such material, would make it very difficult for all but the best undergraduate students to use this book as an introduction to the material. I suppose the book would work well for students coming to mathematics from other disciplines at the graduate level, but I am not sure I think this target audience is large enough to justify another book on the already crowded bookshelves filled with introductory Algebra textbooks. [Darren Glass]

Mathematical finance is a rather new area of research in mathematics, but nevertheless a popular one. This text is just but one example of the current level of sophistication in the application of mathematical techniques to the area of finance. Today not only students and academics would read a book of such caliber but also the practitioners in various financial institutions. I see this book to hold its place quite high with practitioners as bigger financial institutions really do practice the financial instruments explained in the book.

Martingale Method in Financial Modelling is the 2^nd edition of a well-received book. It brings new additions and updates as well as some completely new chapters, such as the one on Volatility Risk. The authors did an excellent job of providing updated references to current work in each of the areas presented in the book, so that the more curious reader can find even more excitement outside of this book.

One important claim of this (quite large) book is that, as the authors mention in the preface, it does not require any knowledge of finance. How true this is really depends on whether you are a mathematician who is interested in financial modeling (and just beginning to investigate problems in finance) or a mathematician with some finance background. Nevertheless, authors do explain financial instruments to a certain extent in the first chapter, and then later on in each chapter as needed. However, I would recommend that one should first read John Hull’s Options, Futures, and Other Derivatives or an equivalent in order to really get a grasp on all these different financial instruments and how they are traded or used. There are almost no examples in the text except some tedious ones in the first chapters, which is why it is even more imperative that the audience has some knowledge of finance in order to have the clear picture in the head of all the possibilities that can occur when modeling financial instrument. For a good experience in reading this book one should have a good knowledge of probability and stochastic calculus. Authors do provide an appendix that covers some stochastic calculus.

The book is divided into two parts: Part I deals with Spots and Futures Markets and Part II with Fixed-income Markets. It starts lightly, trying to make the reader feel comfortable in her chair. Chapter 1 and 2 give preliminaries on financial instruments and discrete time modeling based on arbitrage framework. Chapter 3 jumps on to continuous time and the famous Black-Scholes model. Also we go back to 1900s and the famous Bachelier option pricing formula. Chapter 4 provides valuation formulas for different kinds of foreign currency and equity options. Chapter 5 covers the “flexible” American options in detail and then continues onto exotic options in chapter 6. The issue of modeling volatility has been a big subject over the years. The authors decided, in this new edition, to add a chapter that deals with problems of modelling volatility risk. Chapter 7 deals with implied volatilities, local volatility models as well as stochastic volatility models accompanied by the presentation of the dynamics of volatility surfaces. Part I ends with continuous time security markets covering basic results in a continuous time setting.

Part II is devoted to fixed income markets. This is also a great area of mathematical finance. Naturally, the section starts the reader off with a discussion of interest rates and related concepts. Chapter 9 gives fast descriptions of such fixed income structures and instruments as zero-coupon bonds, forward interest rates, FRAs, interest rate futures, etc. while the chapter finishes off with the presentation of the stochastic models of bond prices. Chapter 10 gives short-term rate models such as single-factor and multi-factor models. Heath, Jarrow and Morton, i.e. HJM approach to term structure modelling is presented in chapter 11. Chapter 12 continues onto arbitrage-free market LIBOR models. The book finishes off with modelling derivatives securities presented in at least two different markets or economies, i.e. the cross-currency derivatives.

Martingale Methods in Financial Modelling is an authoritative text which gives a lot of insight into financial instruments and current modeling practices. The book based on lecture notes and one of its purposes is to serve as a classroom text. Certainly there is plenty of material for a course; however, there are no exercises at the end of the chapters, which makes it hard for students to test their knowledge or to practice the methods learned.

At first glance the book gives an impression of a dry and cold presentation of financial modeling. However, I was glad to find that I was wrong about this. The book is not a dry text, although there are plenty of definitions, theorems and proofs. What makes it come alive is the general understanding of financial instruments and their application. For a graduate student in mathematics this text should not represent too much of a challenge (assuming they have the required background); for practitioners in finance its accessibility will depend only on their mathematics background, as it is written at a sophisticated level.

In general, lots of patience and eagerness to learn the material will get you through a book of this size and content. [Ita Cirovic Donev]

Publication Data

Exploratory Galois Theory, by John Swallow. Cambridge University Press, 2004. Paperback, 220 pp., $34.99. ISBN 0-521-54499-8. Also available in hardcover: $70.00, ISBN 0-521-83650-6.

A First Graduate Course in Abstract Algebra, by W. J. Wickless. Pure and Applied Mathematics 266. Marcel Dekker (now Chapman&Hall/CRC), 2004. Hardcover, 234 pp., $85.00. ISBN 0-8247-5627-4.

Martingale Methods in Financial Modeling, second edition, by Marek Musiela and Marek Rutkowski. Springer-Verlag, 2005. Hardcover, 636 pp., $84.95. ISBN 3-540-20966-2.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

Alvaro Lozano Robledo is a post-doc at Cornell University.

Darren Glass teaches at Gettysburg College.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.

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